Advanced Op Amp Tutorial
This article will
explain advanced op amp behaviour including open
loop gain, closed loop gain, loop gain, phase margin
and gain margin. It expands on the (often incorrect)
assumptions made about op amps that are only
accurate at dc. The text includes simulations in
LTspice^{®}. If you are new to
LTspice, tutorials can be
found on this website.
Operational Amplifiers (Op Amps) are the cornerstone
of analogue electronics. At low frequencies, the
concepts of how an op amp works are very simple and
their circuits are easy to analyse. However, the
basics that are taught in most high schools only
extend to the op amp’s performance at DC. At higher
frequencies the basics are often not applicable and
trying to analyse an AC circuit with DC design rules
often leads to confusion.
This article will spend only the briefest of time
looking at the DC characteristics of op amps then go
on to explain how these characteristics change with
increasing frequency.
The Ideal Op Amp
Text books teach that the ideal op amp has the
following characteristics:
Infinite input impedance
Zero output impedance
Zero dc input offset voltage
Infinite gain
Infinite bandwidth
While not every op amp has high bandwidth, infinite
input impedance, zero output impedance and zero dc
input offset voltage, it is fairly easy to find an
op amp that will come close to these needs for a
particular circuit application. However, no op amps
have infinite gain or bandwidth and in fact the gain
rolls off at very low frequencies and this has an
effect on the assumptions made about the ideal op
amp
The Op Amp at DC
A simple op amp circuit is shown in FIG 1. This has
a non inverting gain of 10.
FIG 1
An LTspice version of this circuit can be downloaded
here: Non
Inverting Op Amp.
The gain of a non inverting op amp is given by
In FIG 1, RF is 9k and RI is 1k, so applying a 10mV
peak input voltage to the non inverting terminal of
FIG 1 implies a voltage of 10x that appears at the
output, i.e. 100mV
Alternatively, if the 2 input terminals regulate to
the same voltage, this creates a current of 10uA
through R2. This current can only come from the
output (as the input terminals do not source
current) meaning that R1 needs to develop a voltage
of 90mV, meaning the output voltage will be at 90mV
+ 10mV = 100mV.
Looking at this circuit another way, there is a
potential divider from the output back to the
inverting input. If the circuit regulates to keep
the 2 inputs the same, then
so
So there are a number of ways of determining the
gain of an op amp.
Now, it is always assumed that the two input
terminals are at the same voltage (ignoring the dc
offset voltage). In fact the voltage across the
input terminals is made up of two components: the dc
offset voltage and a much smaller component that is
dependent on the open loop gain of the amplifier and
it is this second component that most people ignore
which leads to confusion when analysing the op amp
at ac.
The Op Amp at AC Frequencies
FIG 2
In the following
paragraphs, for the sake of clarity, we will assume
that the input offset voltage of the amplifier is
zero. The open loop gain of the
amplifier is equal to the output voltage divided by
the differential voltage across the 2 inputs (in FIG
2 this is the voltage at the OUT node divided by the
voltage Vdiff). The closed loop gain is equal
to the voltage at the OUT node divided by the
voltage at the IN node, as discussed above.
Regardless of the circuit configuration, the op amp
always operates in open loop gain. As circuit
designers, we choose to put components around the
amplifier to give us a certain closed loop gain,
but the amplifier always tries to amplify the
voltage Vdiff by its open loop gain to give a
voltage at the OUT node.
Another way of looking at this is that for any given
voltage at the OUT node, there will be a very small
voltage, Vdiff, across the input nodes whose
magnitude is equal to V(OUT) divided by the open
loop gain. In op amp theory taught in school, the
open loop gain is assumed to be infinite, so the
differential voltage, Vdiff is then infinitely small
(zero). As long as the open loop gain of the
amplifier remains high, this voltage is much smaller
than the input voltage and can be ignored. However,
if the open loop gain of the amplifier goes down,
this voltage starts to get bigger and this is
discussed below.
The open loop gain is assumed to be infinite and
although it is very high at dc, it rolls off soon
after DC and this affects the AC performance of the
op amp. The open loop characteristic of the LT1012
is shown in FIG 3a
FIG 3a
The LTspice plot of this is shown in FIG 3b with the
solid green line showing the gain and the dotted
green line showing the phase. The LTspice circuit
can be downloaded here:
Open Loop Op Amp
Characteristics
FIG 3b
At frequencies below about 0.3Hz the open loop gain
is high at about 126dB (about 2 million). Beyond
0.3Hz, the open loop gain starts to roll of at 20dB
per decade increase in frequency meaning that the
open loop gain goes down by a factor of 10 for every
tenfold increase in frequency. This roll off is
exactly the same as a simple RC filter with a cut
off frequency at 0.3Hz where the response decays at
20dB per decade above the cut off frequency.
If
then to maintain a certain output voltage at V(OUT),
if the open loop gain starts to decrease, the input
voltage Vdiff has to increase.
At low frequencies the voltage Vdiff in FIG 2 will
be small due to the high open loop gain of the op
amp. However, at higher frequencies (above 0.3Hz),
the voltage Vdiff gets bigger and bigger as the open
loop gain gets smaller and smaller.
In FIG 4a a 10mV signal is applied to the circuit in
FIG 2 and a 100mV signal appears at the output, so
we have a gain of 10 as expected. With an input
frequency of 0.01Hz the differential voltage, Vdiff,
measured across the input is 52nV. We can see from
FIG 3a that the open loop gain of the amplifier at
0.01Hz is approximately 126dB (2 million) thus Vdiff
should be 100mV divided by 2 million (50nV) which it
is.
FIG 4a
In FIG 4b, the frequency is increased to 1Hz and all
other circuit parameters remain unchanged. From FIG
3a, we can see that the open loop gain of the op amp
is about 115dB (about 550,000). This is slightly
easier to see in FIG 3b. The differential voltage,
Vdiff, measured across the input is now 182nV. This
corresponds to the output voltage (100mV) divided by
the open loop gain at 1Hz (550,000). Notice also
that in FIG 4a both waveforms are in phase whereas
in FIG 4b there is a phase shift between the input
and the output.
FIG 4b
In FIG 4c, the frequency is increased to 100Hz. The
open loop gain of the op amp at 100Hz is 75dB
(5700). The differential voltage measured across the
input is 17.5uV which corresponds to the output
voltage (100mV) divided by the open loop gain at
100Hz (5700). The output voltage is also phase
shifted with respect to the input.
FIG 4c
Just for completion, changing the feedback resistor
in FIG 2 from 9k to 99k gives the amplifier a gain
of 100. FIG 4d shows the effect on the output and
the differential voltage.
FIG 4d
The output has increased by a factor of 10 (as
expected), but so has the differential voltage. This
is to be expected since the op amp’s open loop gain
remains unchanged at a given frequency. If the
output voltage increases by a factor of 10, for a
given open loop gain, this means the differential
input voltage also has to increase by a factor of
10. This will have important implications and these
will be explained later.
It is interesting to note that although the LT1012
has a very low offset voltage, the simulation model
appears to have a virtually zero dc input offset
voltage which makes our analysis much easier.
Thus it can be seen that the differential input
voltage increases with decreasing open loop gain and
the output undergoes a phase shift above 0.3Hz which
is the frequency at which the open loop gain starts
to roll off (the break frequency).
It should also be noted that, like a simple RC
filter, the phase shift occurs at frequencies around
the break frequency. For a single order RC filter
(one where the gain falls at 20dB per decade) the
phase shift will only ever get to 90 degrees. FIG 4c
shows a phase shift of 90 degrees and if we were to
increase the frequency above the 100Hz shown in FIG
4c, the phase shift would stay at 90 degrees.
We can see from FIG 3 that the slope of the open
loop gain changes above 1MHz and starts to decay at
more than 20dB per decade. This effect is similar to
a second RC filter with a break frequency of 1MHz.
This second RC filter will introduce a further 90
degrees phase shift resulting in a 180 degrees phase
shift at frequencies well in excess of 1MHz.
It is important to remember that although Vdiff is
phase shifted with respect to the output, its
amplitude is only very small compared to the input
voltage of 10mV. The voltage at both input terminals
is still approximately 10mV and as long as Vdiff is
small compared to the input voltage, we need not
worry too much about the phase shift.
Phase Shift
The LT1012’s open loop gain has the frequency
response of a single order low pass filter. A
similar low pass filter (this time with a break
frequency of 1kHz) is shown in FIG 5.
FIG 5a
An LTspice simulation of this circuit can be
downloaded here: First Order
Low Pass Filter.
This filter’s frequency response can be seen in FIG
5b. The break frequency occurs when the output is
3dB lower than the input as represented by the solid
green line below and read off the left hand axis.
FIG 5b
For a single order low pass filter the phase shift
at the break frequency is 45 degrees, as represented
by the dotted green line above and read off the
right hand axis. As an approximation, for a first
order low pass filter, the phase shift at 10x less
than the break frequency is about 0 degrees and at
10x greater than the break frequency, the phase
shift is 90 degrees and this can be seen in FIG 5b.
A mathematical derivation of the amplitude and phase
shift can be seen here:
Low Pass Filter Amplitude and Phase Shift
A low pass filter has a phase lag response.
This means the output voltage reaches its peak after
the input voltage and this can be seen in FIG4c
above, although it is not immediately obvious. The
voltage at the non inverting input is the forcing
function so is at zero phase. The signal passes
through the op amp and undergoes a phase lag and
this appears at the inverting input. The blue
waveform in FIG 4c is measured from the non
inverting terminal to the inverting terminal and
clearly leads the green waveform. Therefore if the
blue waveform were measured from the inverting
terminal to the non inverting terminal it would be
lagging the green waveform.
FIG 6 shows a similar 2 pole filter with one pole at
1kHz and one at 100kHz. It can be seen from FIG 7
that the first pole introduces a roll of at 20 dB
per decade and worst case 90 degree phase shift as
expected and the second pole introduces a further 20
dB per decade roll off and another 90 degree phase
shift.
At the first break frequency the phase shift will be
45 degrees and at the second, it will be 135 degrees
(90 degrees + 45 degrees).
FIG 6
FIG 7
Some amplifiers, including the LT1012 exhibit an
open loop characteristic with 2 break frequencies
(similar to that in FIG 7). With the LT1012, the
first break frequency is at 0.3Hz so introduces a 45
degree phase shift at 0.3Hz (and a 90 degrees phase
shift at frequencies above 3Hz) and the second break
frequency is at 1MHz, at which point the open loop
characteristic will have a 135 degrees phase shift.
The open loop phase shift will tend towards 180
degrees as the frequency gets towards 10MHz.
Loop Gain
A general feedback system, like most op amp
circuits, is represented in FIG 8.
FIG 8
The ‘+’ and ‘‘ inputs represent the non inverting
and inverting inputs to the op amp. The gain block A_{0}
represents the open loop gain of the amplifier and
β
is the fraction of the output fed back (via the
feedback and input resistors). It can be seen that
so
so
so
If A_{0} is large then the overall closed
loop gain approximates to
since
βA_{0}
is large compared with ‘1’ and the A_{0} on
the numerator and denominator cancel. Therefore the
gain of the system approximates to the reciprocal of
the feedback fraction (β).
This can be seen in FIG 1. The feedback fraction is
a simple resistive divider represented by
so the overall gain is 10.
We are now going to introduce the concept of Loop
Gain. It should be noted that loop gain,
open loop gain and closed loop gain
are 3 different parameters and should not be
confused. Loop gain is not something that is
measured in everyday electronics, but it is useful
in explaining how op amps might start misbehaving at
high frequencies.
Referring to FIG 2 we know that open loop gain
is defined as:
And closed loop gain is defined as
We are now going to define loop gain
as the open loop gain, A_{0}, multiplied by
the feedback fraction, β i.e. the gain going around
the loop. This is easy to picture in FIG 8.
If the loop gain is equal to βA_{0} and we
know that the closed loop gain approximates to 1/β,
we can say that the loop gain approximates to the
Open Loop Gain divided by the Closed Loop Gain (i.e.
βA_{0} is equal to A_{0} divided by
1/β) , which is equal to
In other words, the loop gain is a measure of how
big the input voltage, V(IN), is compared with the
differential voltage, Vdiff. However, this is only
an approximation.
To find the exact value of the loop gain, we need to
examine FIG 8. Looking at FIG 8, Vin – βVout is
actually the same as Vdiff in FIG 2 (the
differential voltage across the 2 op amp inputs).
Since the loop gain is the feedback fraction (β)
multiplied by the open loop gain (A_{0}) we
can see from FIG 8 and FIG 2 that the exact
value of loop gain is the voltage at the inverting
terminal, not V(IN), divided by the
differential voltage, Vdiff.
We have seen from the plots of FIG 4 that the
voltage difference between the inverting terminal
and the non inverting terminal gets bigger as
frequency increases implying that our approximation
of the loop gain equalling V(IN)/Vdiff is only
accurate when Vdiff is small.
We would not normally measure the magnitude of the
input voltage and compare it with the differential
voltage, so why is this of any use?
Well, we can see in FIG 4ad that if the voltage
Vdiff is small compared with V(IN) then it presents
no problem and can be ignored. From the equation
above, this represents a high loop gain.
However, if Vdiff starts to become comparable to
V(IN) (as loop gain reduces) it will start to
interfere with the input signal and can no longer be
ignored.
We have already seen in FIG 3 and FIG 5b that if the
op amp has a first order response (i.e. the open
loop gain rolls off at 20dB per decade and the phase
shift is no higher than 90 degrees) that this is not
a problem. However, if the op amp has a second order
response as shown in FIG 7 then it is possible that
the phase of Vdiff can be close to 180 degrees out of phase
with the input at high frequencies. Again this is
not normally a problem if Vdiff is small compared
with V(IN), but if Vdiff is comparable in magnitude
with V(IN), then we are nearing a point of potential
oscillation.
Put another way, under normal circumstances the
voltage fed back is applied to the inverting input
so opposes the input signal. However, if this
voltage is inverted through the amplifier (in FIG 8
this is the A_{0} block), the feedback
signal appears in phase with the input signal and
there is a potential for oscillation.
Plotting the closed loop gain on the graph shown in
FIG 3, we get the graph of FIG 9 with the closed
loop gain shown in red.
FIG 9a
We can also see this plotted in LTspice in FIG 9b
FIG 9b
The Open Loop Gain is at roughly 125dB and is
represented by A_{0} and the Closed Loop
Gain is 20dB and is represented by 1/β
as seen by the red line in the logarithmic plots in
FIG 9a and FIG 9b.
Now, the difference between two numbers on a
logarithmic plot is equal to the ratio of the
two numbers on a linear plot.
We have already established that the Loop Gain
approximates to the Open Loop Gain divided by the
Closed Loop Gain, so on a logarithmic scale, this is
represented by the difference between the
Open Loop Gain plot and the Closed Loop Gain plot
(i.e. the gap between the open loop curve and the
closed loop curve below it) as indicated in FIG 9a.
So the ratio of open loop gain to closed loop gain
is
Thus the gap between the open loop curve and the
closed loop curve is
βA_{0}
which represents the Loop Gain.
βA_{0
}is the loop gain of the system.
Phase Margin and Gain Margin Explained
In FIG 10, consider a signal, Va, applied to the
input of the amplifier, A_{0}. It passes
through the amplifier then through the feedback
network,
β and arrives back at the differential input stage (Vb).
At this point it is inverted by the differential
stage. If Va is subjected to a 180 degrees phase
shift when passing through the gain stage A_{0}
and then inverted by the differential stage, it will
now be back in phase with the original signal.
FIG 10
If at a given frequency the amplitude of Vb is
greater than or equal to the amplitude of Va, then
there is a possibility that the system will
oscillate. Under these conditions, the system needs
no external voltage at Vin to produce a sustained
voltage at Va and Vb . Looking at FIG 10, we can see
that Vb is equal to Va x βA_{0}, where βA_{0
}
is the loop gain of the system, so we can now
see why loop gain is important in determining the
stability of a feedback system. If βA_{0 }
has a phase shift of 180 degrees and a magnitude of
greater than 1 the circuit will oscillate.
This can be
related back to the equation
If the
loop gain,
βA_{0}
has a phase shift of 180 degrees (i.e. is negative)
and has reduced to unity at a certain frequency, the
denominator of the above equation reduces to zero
and the circuit will oscillate at that frequency.
For
the loop gain to equal unity, the Closed Loop Gain
equals the Open Loop Gain, since the loop gain is
defined by Open Loop Gain divided by Closed Loop
Gain. It can be seen that the stability of a system
can thus be determined by looking at the point where
the Open Loop Gain and Closed Loop Gain meet. This
is where the loop gain equals unity.
FIG 11a shows the open loop response of anther op
amp, the LT1226. It can be seen that at an open loop
gain of 20dB we have a phase shift of 180 degrees
(where the dotted white line crosses the dotted
green line and reading off the right hand axis).
This occurs at 65MHz. So potentially there could be
a problem using the LT1226 with gains of less than
20dB. FIG 11b shows the circuit – a non inverting
gain of 100 with an input voltage of 10mV. This Open
Loop test circuit can be downloaded here:
LT1226 Open Loop
Circuit
FIG 11a
FIG 11b
The open loop gain of the LT1226 has its first break
point at 6.5kHz (where the roll off is at 20dB per
decade) and a second one at about 30MHz where the
slope changes from 20dB per decade to 40dB per
decade. At frequencies of 10x less than 6.5kHz, the
phase shift approximates to zero and at frequencies
10x higher than 6.5kHz, the phase shift approximates
to 90 degrees. At 6.5MHz the phase shift is 45
degrees.
FIG 12 shows an input signal of 65kHz (10x higher
than the open loop break frequency). Here we can see
that the differential voltage between the op amp
inputs is phase shifted by 90 degrees with respect
to the output voltage. Remember that the open
loop gain is V(OUT)/Vdiff, so the comparison we
are making in FIG 12 should correspond to the phase
shift of the open loop gain. Indeed FIG 11a shows
the open loop gain of the LT1226 and we can see that
at 65kHz the phase shift is indeed 90 degrees.
This circuit can be downloaded here:
Non Inverting
LT1226 Circuit
FIG 12
We can see that the voltage Vdiff (65uV) is
small compared to the 10mV input signal, so the
output signal is at the amplitude we expect (1V),
for a circuit with a gain of 100.
Increasing the frequency by a factor of 1000 to
65MHz, the open loop gain is 21dB (about 11)
and the phase shift is 180 degrees, from FIG 11a.
Inputting a signal of 65MHz gives the results of FIG
13
FIG 13
We now have a phase shift of 180 degrees of Vdiff
with respect to the output. So why is the circuit
not oscillating? For a circuit to oscillate, we need
a phase shift of 180 degrees around the loop as well
as gain. We have the phase shift, but do we have
enough gain? We now need to look at the signal at
each of the input terminals to see if there is
enough gain in the loop. So far we have looked at
the voltage across the 2 op amp inputs (Vdiff),
because it is easier to measure a Vdiff of, say,
100nV rather than measure the input voltage of 10mV
and the voltage at the inverting terminal which is
(10mV – 100nV). To get a true idea of instability we
have to measure the voltage at each input.
FIG 14
FIG 14 shows the voltage at the inverting terminal
(V(n001)), the input and the output. The input is at
10mV, but the output voltage has suffered because of
the poor open loop gain of the amplifier. Indeed
measuring the voltage at the output pin and dividing
it by the differential voltage across the inputs, we
should still arrive at the open loop gain figure of
approximately 21dB. It should be noted that there is
a 40mV output offset voltage that should be
removed from the output voltage reading before
calculating the gain.
Looking at the voltage at the inverting terminal, we
can see that it is one hundredth of the voltage at
the output and it is in phase with the output
voltage. This is to be expected as there is a
resistive divider (that gives zero phase shift) from
the output back to the inverting terminal made up of
resistors R2 and R3 and giving an attenuation of 100
as seen in FIG 11b.
We can also see that the voltage at the inverting
terminal (V(n001)) is substantially lower than the
input voltage, so although it is out of phase with
the input by 180 degrees, it is not bigger than V(IN)
so the system cannot oscillate.
Now, we can reduce the gain of the amplifier by
reducing, for example, the feedback resistor R2 in
FIG 11b. In doing so the attenuation effect of the
resistors R2 and R3 gets less, so more voltage
appears at the inverting input. Thus it can now be
seen that at a given frequency where the voltage at
the inverting terminal is inverted with respect to
the voltage at the non inverting terminal,
decreasing the gain is not a good idea, as this will
increase the voltage at the inverting terminal.
Ultimately we will reach a point where the voltage
at the inverting terminal is 180 degrees out of
phase and greater in amplitude than the voltage at
the non inverting terminal. We now have a condition
for oscillation. This is why many op amps have a
minimum gain stability. If the gain is reduced below
this point, the op amp will start to oscillate. This
can be seen in FIG 15.
FIG 15
Here R2 has been reduced to 1k and R3 kept at 100
Ohms and this has caused an increase in the voltage
at the inverting input. This voltage is higher in
amplitude than the 10mV input signal, so we can
remove the 10mV input signal and the circuit will
continue to oscillate. This can be seen in FIG 15.
(The input frequency has actually been reduced to
1Hz).
The output voltage has a peak to peak amplitude of
1.049V and the inverting input has an amplitude of
94.67mV. The feedback fraction is 100/(100+1000) =
0.091. If the non inverting terminal is at 0V, then
the differential voltage is at 94.67mV. We can work
out the open loop gain to be 1.049V/94.67mV = 11.08.
Thus the feedback fraction multiplied by the open
loop gain is 0.091 x 11.29 = 1.01. Thus we have 180
degrees phase shift with a loop gain of >1 and these
are the conditions for possible oscillation.
Gain and phase margin are a measure of how close to
the point of oscillation the circuit is. In other
words how close to 180 degrees phase shift or unity
gain the loop gain is. It is a measure of the loop
gain of the circuit, not the closed loop gain
or the open loop gain.
Phase margin
is a measure of how close the loop gain is to
having 180 degrees of phase shift when the loop gain
is unity. If
βA_{0}
has 180 degrees of phase shift when it has a
magnitude of unity, the circuit has zero degrees of
phase margin and will oscillate. If the loop gain
has a phase shift of 160 degrees, the circuit has a
phase margin of 20 degrees.
Gain margin
is a measure of how far below unity the
loop gain of the circuit gain is when the loop
gain,
βA_{0}, has a phase shift of
180 degrees. If the loop gain has a phase shift of
180 degrees and a loop gain of 0.6, the circuit has
a gain margin of 0.4. A circuit with a loop gain of
0.8 has a gain margin of 0.2 and hence is nearer to
the point of oscillation.
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registered trademark of Linear Technology
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